Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 98   | 1 | 2 |

Section Agarwal's FFT implementation of the Fourier method

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Agarwal's FFT implementation of the Fourier method

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Agarwal (1978)[link] rederived and completed Cruickshank's results at a time when the availability of the FFT algorithm made the Fourier method of calculating the coefficients of the normal equations much more economical than the standard method, especially for macromolecules.

As obtained by Cruickshank, the modified Fourier method required a full 3D Fourier synthesis

  • – for each type of parameter, since this determines [via the polynomial [P_{p} ({\bf h})]] the type of differential synthesis to be computed;

  • – for each type of atom [j \in J], since the coefficients of the differential synthesis must be multiplied by [g_{j}({\bf h})].

Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a real-space convolution between the differential synthesis and [\sigma\llap{$-$}_{j} ({\bf x})], the standard electron density [\rho\llap{$-\!$}_{j}] for atom type j (Section[link]) smeared by the isotropic thermal agitation of that atom. Since [\sigma\llap{$-$}_{j}] is localized, this convolution involves only a small number of grid points. The requirement of a distinct differential synthesis for each parameter type, however, continued to hold, and created some difficulties at the FFT level because the symmetries of differential syntheses are more complex than ordinary space-group symmetries. Jack & Levitt (1978)[link] sought to avoid the calculation of difference syntheses by using instead finite differences calculated from ordinary Fourier or difference Fourier maps.

In spite of its complication, this return to the Fourier implementation of the least-squares method led to spectacular increases in speed (Isaacs & Agarwal, 1978[link]; Agarwal, 1980[link]; Baker & Dodson, 1980[link]) and quickly gained general acceptance (Dodson, 1981[link]; Isaacs, 1982a[link],b[link], 1984[link]).


Agarwal, R. C. (1978). A new least-squares refinement technique based on the fast Fourier transform algorithm. Acta Cryst. A34, 791–809.
Agarwal, R. C. (1980). The refinement of crystal structure by fast Fourier least-squares. In Computing in Crystallography, edited by R. Diamond, S. Ramaseshan & K. Venkatesan, pp. 18.01–18.13. Bangalore: The Indian Academy of Science.
Baker, E. N. & Dodson, E. J. (1980). Crystallographic refinement of the structure of actinidin at 1.7 Å resolution by fast Fourier least-squares methods. Acta Cryst. A36, 559–572.
Dodson, E. J. (1981). Block diagonal least squares refinement using fast Fourier techniques. In Refinement of Protein Structures, compiled by P. A. Machin J. W. Campbell & M. Elder (ref. DL/SCI/R16), pp. 29–39. Warrington: SERC Daresbury Laboratory.
Isaacs, N. W. (1982a). The refinement of macromolecules. In Computational Crystallography, edited by D. Sayre, pp. 381–397. New York: Oxford University Press.
Isaacs, N. W. (1982b). Refinement techniques: use of the FFT. In Computational Crystallography, edited by D. Sayre, pp. 398–408. New York: Oxford University Press.
Isaacs, N. W. (1984). Refinement using the fast-Fourier transform least squares algorithm. In Methods and Applications in Crystallographic Computing, edited by S. R. Hall & T. Ashida, pp. 193–205. New York: Oxford University Press.
Isaacs, N. W. & Agarwal, R. C. (1978). Experience with fast Fourier least squares in the refinement of the crystal structure of rhombohedral 2-zinc insulin at 1.5 Å resolution. Acta Cryst. A34, 782–791.
Jack, A. & Levitt, M. (1978). Refinement of large structures by simultaneous minimization of energy and R factor. Acta Cryst. A34, 931–935.

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